Two-Variable Quantitative Data • Most statistical studies involve more than one variable. • We may believe that some of the variables explain or even cause changes in the variables. Then we have explanatory and response variables. • Explanatory—like the independent variable, it attempts to explain the observed outcomes. • Response—like the dependent variable, it measures an outcome of a study.
Examples Identify the explanatory and response variables: • Alcohol causes a drop in body temperature. To measure this, researchers give several different amounts of alcohol to mice, then measure the change in their body temperature after 15 minutes. • If an object is dropped from a height, then its downward speed theoretically increases over time due to the pull of gravity. To test this, a ball is dropped and at certain intervals of time, the speed of the ball is measured.
Scatterplots • Used for two-variable quantitative data! • Explanatory variable goes on the x-axis • Response variable goes on the y-axis • The explanatory variable does not necessarily “CAUSE” the change in the response variable.
Displaying Relationships: Scatterplots Make a scatterplot of the relationship between body weight and pack weight. Since Body weight is our eXplanatory variable, be sure to place it on the X-axis! Scatterplots and Correlation
In sentence form… There is a (strong/weak), (positive/negative), (linear/non-linear) relationship between (your two variables).
Interpreting Scatterplots Scatterplots and Correlation Outlier • There is one possible outlier, the hiker with the body weight of 187 pounds seems to be carrying relatively less weight than are the other group members. Strength Direction Form • There is a moderately strong, positive, linear relationship between body weight and pack weight. • It appears that lighter students are carrying lighter backpacks.
Adding Categorical Variables to Scatterplots • You can use different plotting symbols or different colors to designate a categorical variable. • You still have two quantitative variables, but you can add a “category” to these variables. • See p. 179 for an example
Some quick tips for drawing scatterplots • Choose an appropriate scale for the axes. Use a break if appropriate. • Label, Label, Label… • If you are given a grid, try to use a scale that will make the scatterplot use the whole grid.
Section 3.2 CorrelationWe are not good judges! • We shouldn’t just rely on our eyes to tell us how strong a linear relationship is. • We have a numerical indication for how strong that linear relationship is – it’s called CORRELATION.
Scatterplots and Correlation • Definition: • The correlation r measures the strength of the linear relationship between two quantitative variables. • r is always a number between -1 and 1 • r > 0 indicates a positive association. • r < 0 indicates a negative association. • Values of r near 0 indicate a very weak linear relationship. • The strength of the linear relationship increases as r moves away from 0 towards -1 or 1. • The extreme values r = -1 and r = 1 occur only in the case of a perfect linear relationship.
Facts About Correlation • It does not require a response and explanatory variable. Ex. How are SAT math and verbal scores related? • If you switch the x and the y variables, the correlation doesn’t change. • If you change the units of measurement for x and/or y, the correlation doesn’t change. • Positive r values indicate a positive relationship; negative values indicate a negative relationship. Remember… not cause.
More Facts • Correlation measures the strength of the LINEAR relationship. It doesn’t measure curved relationships. • Correlation is strongly affected by outliers. • r does not have a unit.
Homework 3.3, 3.6, 3.7, 3.10, 3.14, 3.20